A note on simplicial depth function

We investigate the behaviour of simplicial depth under the perturbation (1−ε)F+ε δ _z , where F is a p-dimensional probability distribution and δ _z is the point-mass distribution concentrated at the point z. The influence function of simplicial depth at the point x, up to a scalar multiplier, turns out to be the difference between the conditional depth, given that one of the vertices of the random simplex is fixed at the position z, and the unconditional depth. The scalar multiplier is p+1, which suggests that simplicial depth can be more sensitive to perturbations as the dimensionality grows higher. The geometrical properties of the influence function give new insight into the observed behaviour of simplicial depth and its relation with halfspace depth. The behaviour of the perturbed simplicial median is also investigated.     

A robust principal component analysis

This work is concerned with robustness in Principal Component Analysis (PCA). The approach, which we adopt here, is to replace the criterion of least squares by another criterion based on a convex and sufficiently differentiable loss function ρ. Using this criterion we propose a robust estimate of the location vector and introduce an orthogonality with respect to (w.r.t.) ρ in order to define the different steps of a PCA. The influence functions of a vector mean and principal vectors are developed in order to provide method for obtaining a robust PCA. The practical procedure is based on an alternative-steps algorithm.     

Weighted quantile regression with nonelliptically structured covariates

Although quantile regression estimators are robust against low leverage observations with atypically large responses (Koenker & Bassett 1978), they can be seriously affected by a few points that deviate from the majority of the sample covariates. This problem can be alleviated by downweighting observations with high leverage. Unfortunately, when the covariates are not elliptically distributed, Mahalanobis distances may not be able to correctly identify atypical points. In this paper the authors discuss the use of weights based on a new leverage measure constructed using Rosenblatt's multivariate transformation which is able to reflect nonelliptical structures in the covariate space. The resulting weighted estimators are consistent, asymptotically normal, and have a bounded influence function. In addition, the authors also discuss a selection criterion for choosing the downweighting scheme. They illustrate their approach with child growth data from Finland. Finally, their simulation studies suggest that this methodology has good finite‐sample properties.     

A software reliability model using mean residual quantile function

In this paper, we propose a class of distributions with the inverse linear mean residual quantile function. The distributional properties of the family of distributions are studied. We then discuss the reliability characteristics of the family of distributions. Some characterizations of the class of distributions are also discussed. The parameters of the class of distributions are estimated using the method of L-moments. The proposed class of distributions is applied to a real data set.     

A new approach to use multivariate auxiliary information in sample surveys

A new approach to form multivariate difference estimator is suggested which does not require the knowledge of unknown population parameters as such. It gives minimum variance among the class of multivariate difference estimators. The performance of this estimator with respect to Des Raj's (J. Amer. Statist. Assoc. 60 (1965), 270–277) multivariate difference estimator is illustrated. Using the information on two auxiliary variates, the robustness of Des Raj's estimator y_d is studied empirically. Two new estimators to estimate population mean/total are developed on the same lines as that of y_d. The performance of these estimators is studied for a wide variety of populations.     

A class of distributions with the quadratic mean residual quantile function

Abstract

The present paper introduces a new family of distributions with quadratic mean residual quantile function. Various distributional properties as well as reliability characteristics are discussed. Some characterizations of the class of distributions are presented. The estimation of parameters of the model using method of L-moments is studied. The practical application of the class of models is illustrated with a real life data set.     

A regression approach for estimating the parameters of the covariance function of a stationary spatial random process

We consider the problem of estimating the parameters of the covariance function of a stationary spatial random process. In spatial statistics, there are widely used parametric forms for the covariance functions, and various methods for estimating the parameters have been proposed in the literature. We develop a method for estimating the parameters of the covariance function that is based on a regression approach. Our method utilizes pairs of observations whose distances are closest to a value h>0$h>0$ which is chosen in a way that the estimated correlation at distance h is a predetermined value. We demonstrate the effectiveness of our procedure by simulation studies and an application to a water pH data set. Simulation studies show that our method outperforms all well-known least squares-based approaches to the variogram estimation and is comparable to the maximum likelihood estimation of the parameters of the covariance function. We also show that under a mixing condition on the random field, the proposed estimator is consistent for standard one parameter models for stationary correlation functions.     

Reduced Palm distribution generating function for a spatial point process

The generating function of a marginal distribution of the reduced Palm distribution of a spatial point process is considered. It serves as a bivariate summary function, providing more information than some other popular univariate summary functions, such as the reduced second-moment function and the nearest-neighbour distance distribution function. Simulation confirmed that the new summary function is more informative when applied to patterns that exhibit both clustering and regularity on the same scale of observation.     

Some new stochastic orders based on quantile function

Recently, in the literature, the use of quantile functions in the place of distribution functions has provided new models, alternative methodology and easier algebraic manipulations. In this paper, we introduce new orders among the random variables in terms of their quantile functions like the reversed hazard quantile function, the reversed mean residual quantile function and the reversed variance residual quantile function orders. The relationships among the proposed orders and some existing orders are also discussed.     

A New Conjoint Analysis Procedure with Application to Marketing Research

Abstract

In this article we propose some extensions and applications of the nonparametric combination of dependent rankings (see Pesarin, F., Lago, A. (2000). Nonparametric combination of department rankings with applications to the quality assessment of industrial products. Metron LVIII (1–2):39–52.) This methodology is applied to Conjoint Analysis in order to aggregate (ex ante) preferences from a group of individuals. Furthermore, a new global association test (GAT) is introduced in order to test for the association of the global ranking with all attributes of interest. The GAT procedure allows the experimenter to have clear indications on significant attributes by considering the intensity of the optimal weights given by the procedure itself. This may help the experimenter in interpreting the usual analysis involving the normal plot for detecting active effects.     

A weighted spatial median for clustered data

A weighted spatial median is proposed for the multivariate one-sample location problem with clustered data. Its limiting distribution is derived under mild conditions (no moment assumptions) and it is shown to be multivariate normal. Asymptotic as well as finite sample efficiencies and breakdown properties are considered, and the theoretical results are supplied with illustrative examples. It turns out that there is a potential for meaningful gains in estimation efficiency: the weighted spatial median has superior efficiency to the unweighted spatial median particularly when the cluster sizes are widely disparate and in the presence of strong intracluster correlation. The unweighted spatial median for clustered data was considered earlier by Nevalainen et al. (Can J Statist, in press, 2007). The proposed weighted estimators provide companion estimates to the weighted affine invariant sign test proposed recently by Larocque et al. (Biometrika, in press, 2007). An affine equivariant weighted spatial median is discussed in parallel.     

Biased Bootstrap Methods for Reducing the Effects of Contamination

Contamination of a sampled distribution, for example by a heavy-tailed distribution, can degrade the performance of a statistical estimator. We suggest a general approach to alleviating this problem, using a version of the weighted bootstrap. The idea is to 'tilt' away from the contaminated distribution by a given (but arbitrary) amount, in a direction that minimizes a measure of the new distribution's dispersion. This theoretical proposal has a simple empirical version, which results in each data value being assigned a weight according to an assessment of its influence on dispersion. Importantly, distance can be measured directly in terms of the likely level of contamination, without reference to an empirical measure of scale. This makes the procedure particularly attractive for use in multivariate problems. It has several forms, depending on the definitions taken for dispersion and for distance between distributions. Examples of dispersion measures include variance and generalizations based on high order moments. Practicable measures of the distance between distributions may be based on power divergence, which includes Hellinger and Kullback–Leibler distances. The resulting location estimator has a smooth, redescending influence curve and appears to avoid computational difficulties that are typically associated with redescending estimators. Its breakdown point can be located at any desired value ε∈ (0, ½) simply by 'trimming' to a known distance (depending only on ε and the choice of distance measure) from the empirical distribution. The estimator has an affine equivariant multivariate form. Further, the general method is applicable to a range of statistical problems, including regression.     

A simulation study of properties of a regressive logistic model

A regressive logistic model for the analysis of data with dependent binary observations is constructed by successively conditioning on preceding observations. The properties of this model are investigated and compared to those of the ordinary logistic regression model in which the dependence is not considered, using computer simulation. Comparison criteria include the magnitude of the bias and the total mean square error (MSE) of the regression coefficient β and the significance level. The results suggest the regressive model significantly improves the estimation of the regression coefficient     

Testing whether the survival function is multivariate new better than used

A test is proposed to test that a life distribution is multivariate exponential (MVE) against the alternative that it is multivariate new better than used (MNBU) class of alternatives. We also show that the proposed test is consistent for the alternatives of multivariate new better than used in expectations (MNBUE).     

A semi-parametric approach to robust parameter design

Parameter design or robust parameter design (RPD) is an engineering methodology intended as a cost-effective approach for improving the quality of products and processes. The goal of parameter design is to choose the levels of the control variables that optimize a defined quality characteristic. An essential component of RPD involves the assumption of well estimated models for the process mean and variance. Traditionally, the modeling of the mean and variance has been done parametrically. It is often the case, particularly when modeling the variance, that nonparametric techniques are more appropriate due to the nature of the curvature in the underlying function. Most response surface experiments involve sparse data. In sparse data situations with unusual curvature in the underlying function, nonparametric techniques often result in estimates with problematic variation whereas their parametric counterparts may result in estimates with problematic bias. We propose the use of semi-parametric modeling within the robust design setting, combining parametric and nonparametric functions to improve the quality of both mean and variance model estimation. The proposed method will be illustrated with an example and simulations.     

A working estimating equation for spatial count data

This paper proposes a working estimating equation which is computationally easy to use for spatial count data. The proposed estimating equation is a modification of quasi-likelihood estimating equations without the need of correctly specifying the covariance matrix. Under some regularity conditions, we show that the proposed estimator has consistency and asymptotic normality. A simulation comparison also indicates that the proposed method has competitive performance in dealing with over-dispersion data from a parameter-driven model.     

Nonparametric Simultaneous Tests for Location and Scale Testing: A Comparison of Several Methods

The two-sample location-scale problem arises in many situations like climate dynamics, bioinformatics, medicine, and finance. To address this problem, the nonparametric approach is considered because in practice, the normal assumption is often not fulfilled or the observations are too few to rely on the central limit theorem, and moreover outliers, heavy tails and skewness may be possible. In these situations, a nonparametric test is generally more robust and powerful than a parametric test. Various nonparametric tests have been proposed for the two-sample location-scale problem. In particular, we consider tests due to Lepage, Cucconi, Podgor-Gastwirth, Neuhäuser, Zhang, and Murakami. So far all these tests have not been compared. Moreover, for the Neuhäuser test and the Murakami test, the power has not been studied in detail. It is the aim of the article to review and compare these tests for the jointly detection of location and scale changes by means of a very detailed simulation study. It is shown that both the Podgor–Gastwirth test and the computationally simpler Cucconi test are preferable. Two actual examples within the medical context are discussed.     

An automatic classification and robust segmentation procedure of spatial objects

This paper proposes a statistical procedure for the automatic volumetric primitives classification and segmentation of 3D objects surveyed with high density laser scanning range measurements. The procedure is carried out in three main phases: first, a Taylor’s expansion nonparametric model is applied to study the differential local properties of the surface so to classify and identify homogeneous point clusters. Classification is based on the study of the surface Gaussian and mean curvature, computed for each point from estimated differential parameters of the Taylor’s formula extended to second order terms. The geometrical primitives are classified into the following basic types: elliptic, hyperbolic, parabolic and planar. The last phase corresponds to a parametric regression applied to perform a robust segmentation of the various primitives. A Simultaneous AutoRegressive model is applied to define the trend surface for each geometric feature, and a Forward Search procedure puts in evidence outliers or clusters of non stationary data. An erratum to this article can be found at     

THE EFFECTS OF MISCLASSIFICATION COSTS AND SKEWED DISTRIBUTIONS IN TWO-GROUP CLASSIFICATION

ABSTRACT

In this study, Monte Carlo simulation experiments were employed to examine the performance of four statistical two-group classification methods when the data distributions are skewed and misclassification costs are unequal, conditions frequently encountered in business and economic applications. The classification methods studied are linear and quadratic parametric, nearest neighbor and logistic regression methods. It was found that when skewness is moderate, the parametric methods tend to give best results. Depending on the specific data condition, when skewness is high, either the linear parametric, logistic regression, or the nearest-neighbor method gives the best results. When misclassification costs differ widely across groups, the linear parametric method is favored over the other methods for many of the data conditions studied.